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Full text of "Stable and unstable molecules under supercritical and cryogenic conditions"

STABLE AND UNSTABLE MOLECULES UNDER SUPERCRITICAL 
AND CRYOGENIC CONDITIONS 



By 
TROY D. HALVORSEN 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT 

OF THE REQUIREMENTS FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 

UNIVERSITY OF FLORIDA 

1998 



ACKNOWLEDGEMENTS 
The author wishes to thank first and foremost his family whose love and support 
have fostered his growth as an individual and has propelled him to this position in his 
life. His mother (Susan) who personally sacrificed and was unwilling to let the author 
accept anything but the best. His dad (Joe) who first introduced the author to academia 
and has "gently'" guided his path towards higher education. The author does not believe 
that his dad would have ever believed in his wildest dreams that he would eventually 
accept an assistant professorship at Truman State University. When his dad dropped him 
off at Illinois State University in the fall of 1986 to begin his undergraduate career as a 
chemistry major, the author believes that his exact words were "I think your strengths are 
in Sociology." 

The author would also like to acknowledge his wife. Kim. for her love and 
support. Kim almost did not follow the author to Gainesville, but the author cannot 
imagine life without her and their new baby girl, Alana Marie ("Peaches"), who has been 
a divine gift. 

The author is deeply indebted to Dr. William Weltner. Jr.. who unquestioningly 
accepted the author into his group with only two years remaining in his Ph.D. work. The 
author thanks him for giving him the opportunity, and it cannot go without saying that 
Dr. Welter's striving for the truth has made an indelible impression. 



n 



The author would also like to extend his thanks to Dr. Sam Colgate for allowing 
the him to work in his laboratory and for passing on to him his vast knowledge and skills. 
Thank you. 

The author cannot say enough about the wonderful support staff at the University 
of Florida, especially the machine shop personnel. The projects the author has worked on 
would not have come to fruition without the skill and knowledge of Joe Shalosky, Todd 
Prox, and Mike Herlevich. We are truly spoiled to have access to these people. 

Thanks also goes out to the author's cohorts and friends at UF. Dr. Aaron 
Williams. Dr. Heather Weimer, Dr. John Graham. Johnny Evans, and especially Dr. 
Richard Van Zee who might be the single most impressive scientist I have been 
associated with. 

The author cannot forget about the "crue" from Illinois State University ("the 
Illinois mafia") who will always be apart of the author's life; David E. Kage, Dr. Nick 
Kob, Dr. Richard Burton, Dr. Eugene Wagner, Dr. Scott Kassel, and Ben Novak. 

The author must also acknowledge the National Science Foundation (NSF) for 
financial support. 



in 



TABLE OF CONTENTS 

Page 

ACKNOWLEDGEMENTS ii 

ABSTRACT vi 

CHAPTERS 

1 ELECTRON SPIN RESONANCE (ESR) THEORY 1 

Classical Description 1 

Quantum-Mechanical Treatment 2 

2 THE NATURE OF A SPIN PROBE UNDER THE 
INFLUENCE OF SUPERCRITICAL CARBON DIOXIDE 

(COj) 11 

General Description of Supercritical Fluids 11 

Introduction 12 

Experimental 14 

Results and Discussion 18 

Conclusion 23 

3 VIBRATIONAL SPECTROSCOPY THEORY 50 

Classical Description 50 

Quantum-Mechanical Treatment 53 

4 INFRARED SPECTRA OF Nb l2 C, Nb l3 C AND 
NbO : MOLECULES MATRIX ISOLATED IN 

RARE GAS MATRICES 58 

Introduction 58 

Experimental 59 

Results and Discussion 60 



IV 



The Ground State ( 2 A V2 ) Vibrational Properties of Nb 12 C 

andNb 13 C 67 

The NbO : molecule 68 

Conclusion 72 

5 DENSIMETER 74 

Introduction 74 

Thermodynamic Relationships 75 

First Principles 78 

Design of Densimeter 80 

Densimeter Development 84 

Experimental 89 

Results 115 

Conclusion 116 



REFERENCES 122 

BIOGRAPHICAL SKETCH 125 



Abstract of Dissertation Presented to the Graduate School of the University of 
Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 

STABLE AND UNSTABLE MOLECULES UNDER SUPERCRITICAL AND 
CRYOGENIC CONDITIONS 

By 

Troy D. Halvorsen 

August 1998 

Chairman: Professor William Weltner, Jr. 
Major Department: Chemistry 

The electron spin resonance spectra of a spin label (di-tertiary butyl nitroxide, 
DTBN) dissolved in supercritical carbon dioxide were obtained. By investigating the 
band shapes of the spectra, information about the nature of the solvent can be inferred. 
Comparison to simulated spectra provided a means of interpreting these data. 

The spectra of Nb l2 C, Nb l3 C and NbO, in argon, krypton and neon matrices were 
obtained using Fourier Transform infrared spectroscopy (FTIR). The resulting spectra 
were assigned through comparison of the isotopic shifts in vibrational frequencies. 

A prototype high pressure densimeter was designed, built and successfully tested 
with argon and carbon dioxide around ambient temperatures and pressures up to 1800 
psia. 



vi 



CHAPTER 1 
ELECTRON SPIN RESONANCE (ESR) THEORY 

Classical Description 

Classically speaking, an atom may possess electronic orbital motion which creates 

a magnetic dipole (|i). The description of a dipole moment induced by general angular 

momentum is given in equation 1, where p L is the orbital angular momentum, c is the 

velocity of light, and e and m are the elementary charge and fundamental mass of an 

electron, respectively. The dipole moment is therefore proportional to the magnitude and 

h = feK o) 

direction of the orbital angular momentum (p L ) and is scaled by a term more commonly 
referred to as the gyromagnetic ratio (y = -e/2mc). 

The intrinsic magnetic moment associated with an electron regardless of motion 
has been termed '"spin." The dipole moment associated with the spin of an electron is 
represented in equation 2, and it should be noted that the gyromagnetic ratio has an 
additional scaling factor of 2, which is added to account for its anomalous behavior. The 
contributions of spin (S) and orbital angular momenta (L) can couple to give a resultant 
total angular momentum (J) that can interact with an applied or local magnetic field. The 
energy of a magnetic dipole that is "perturbed" by a magnetic field is given classically by 
equations. 



•* = i^rJ Ps (2) 



B = -H • H cosG (3) 

Quantum mechanics has relegated the angle 9 (with respect to the dipole moment 
and magnetic field) to discrete space-fixed orientations. There are 2J + 1 orientations of 
the total angular momentum (J) with projections of nijh along the magnetic field axis, 
where im can assume values of nij = J, J - 1 , ... -J. The angular momentum coincident 
with the field are therefore the integral multiples of p ; = nijh. The dipole moment 

coincident with the magnetic field can then be rewritten as in equation 4. Here the term 
(eh/47tmc = PJ has been defined as the Bohr magneton. Therefore, the Zeeman energies 

^ = -i£^h (4) 

for a magnetic dipole in a magnetic field are given in equation 5, where a quantum 
electrodynamics correction needs to be included in lieu of the constant factor of 2. This 
correction is deemed the g-factor (g e = 2.0023). 

E = -u H • H = 2p> s tf = g e %m s H (5) 

Quantum-Mechanic al Treatment 
The simplest, yet the most rigorous and instructive manner in which to introduce 
the theory of Electron Spin Resonance (ESR) is to investigate the spin system of the 
simplest chemical entity known; the interaction of an electron (fermion) with a proton 
(fermion), routinely known as the ground state hydrogen atom ('H). In this case, the spin 
of the electron ( S = l A ) does not solely produce fine structure, but may also interact with 



3 
the spin of the nucleus ( I = Vi in this case) to give hyperfine structure and therefore 
affords a rich example of this intricate interaction that is common to ESR. Furthermore, 
the Hamiltonian for the spin degrees of freedom of a hydrogen atom is not complicated 
by the anisotropics of the g-tensor and hyperfine interaction tensor (A) due to the lack of 
spin-orbit coupling and the spherical nature of the electronic ground state. Thus, the 
terms needed in the Hamiltonian are represented in equation 6. 

H = g e P e H T • S + A S T ■ i - g n £ n H T • I (6) 



ST e P e -tf r ' S ■ electron Zeeman term 
A S T • I ■ hyperfine interaction term 
g n P n r ■ nuclear Zeeman term 

When the z-axis is chosen to be the axis of space-quantization and coincident with 
the magnetic field (H), the Hamiltonian transforms via equation 7. With the use of the 
general raising and lowering operators (equations 8 and 9). equation 7 converts into 

H = gr e p e tf • S z + A (Sz ■ Iz + S y ■ I y + S x ■ I x ) - g n $ n H ■ f, (7) 

final functional form, which is represented in equation 10. The Hamiltonian that has been 

J + = j x + iJ y (8) 

J. = j x - ij y (9) 

constructed will operate on the four independent basis functions | M s , M x ) for the 



-(s + ■ I. + S. • J + ) 



S, • J. + 

2 



H = gr e p e tf • S z + A z 
~ STnPnH • I, 
hydrogen atom in the manner indicated in equation 1 1 . The evaluation of the matrix 



(10) 



{Ms.Mj \h\ M s ',M r ') (11) 
elements have been worked out previously, 1 and the nonzero elements are contained 
below in equations 12-16. The resultant 4x4 matrix, depicted in Figure 1.1. can be 

(M s |S Z | M s ) = M s (12) 

(m x \l z \ M x ) = M x (13) 

(M s ± 1, M x \S ± I Z \ M s , M T ) = M x [s(S + l) - M S (M S ± l)]' /2 (14) 

(M s ± 1, M x ± \\S ± I ± \ M s , M x ) = 

[S(S + 1) -M S (M S ± I)]' 72 • [1(1 + 1) - M Z (M X ± 1)]' /2 ° 5) 



(16) 



(M a ± 1, M x + 1(5 ± I T | M s , M x ) = 

[S(S + 1) - M S (M S ± 1)]' /2 • [1(1 + 1) - Ml (M s + 1)]' /2 

solved as a secular determinant. To give a lucid representation of its solution, the matrix 
has been block factored along the diagonal. The two (lxl) diagonal elements give the 
following energies (equations 17 and 18), and the remaining (2 x 2) determinant can be 

E a(eMn) = + ~ S^ePe" + - A - - g n $ n H (17) 

S P(e)P(n) = --STePe-tf + ~ A, + -g n P n H (18) 

expanded and solved to produce the energies shown in equations 19 and 20. Note that the 
brackets that surround these particular eigenvalues (equations 19 and 20) indicate that 
they are only true eigenvalues in the limit of high magnetic field. In reality, the two 
states are linear combinations of the eigenstates [a (e) P (n) and P (e) a (n) ] and their "purity" is 



A 

c 
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B {a(e) P (n)} = " \ A ° + \ [&ePe + ST n P n )V + A. 5 

- 1 1 r. 2 ~> 



1/2 



1/2 



(19) 



(20) 



contingent on the strength of the applied magnetic field. For instance, in a weak field the 
states are heavily mixed and strong coupling occurs between the states, which gives rise 
to four possible transitions. 

The above solutions are referred to as the Breit-Rabi energies and are illustrated in 
Figure 1 .2 as a function of low, intermediate, and high magnetic fields. The upper panel 
shows the low field region and the lower panel is at intermediate and high field. The 
Breit-Rabi equation was originally utilized to explain the transitional behavior of 
hydrogen atoms in a low-field molecular beam. 2 

The presence of off-diagonal terms in the matrix shown in Figure 1.1, necessarily 
indicates that the original basis functions are not all eigenfunctions of the Hamiltonian. A 
new basis set can obtained that are eigenfunctions of the Hamiltonian. which is based on 
a coupled representation of the individual angular momenta (see equation 21). This 

F = S + I (21) 

results in a total angular-momentum with a new set of quantum numbers, which can be 

represented in ket notation as | F, M F ) . By inspection of the original matrix (see Figure 
1.1), there are two functions that already exist as eigenfunctions by virtue of being 
diagonal elements ( a (e)/ a (n) ) and | p (e)/ p (n) )), and they can be represented in the 
coupled format as | F, M F ) = | l,+l) and | l,-l). The remaining coupled 




Figure 1.2. The Breit-Rabi diagram for the Hydrogen atom. 



eigenfunctions \ F, M F ) - j 1,0) and | 0,0) can be solved for by the expansion of the 
(2 x2) matrix as mentioned previously and represented as linear combinations below 
(equations 22 and 23). 

The angle co in the equations below determines the weight of hyperfine and 
Zeeman energies and can be extracted from the following relationships (equations 24 and 

| 1,0) = cos co j a (e)l P (n) ) + sin co | p (e) , a (n) ) (22) 

| 0,0) = - sin co | a (e) , p (n) ) + cos co| P (e) , a (n) ) (23) 

25). At zero field, (h = 0, co -> n / 4) , the two states are equally mixed and the 

sin2co - |(1 + ^ 2 )" 1/2 | (24) 

cos2co = S|(l + ^ 2 )" 1/2 | (25) 

where % = (g e p e + gr n p n )tf / A a 
resultant linear combinations are given in equations 26 and 27. At the other extreme, as 
H approaches infinity (h -► «>, co -> 0) the two states collapse into separate and 
distinct states (see equations 28 and 29). 

| 1,0) = ^(|a( e) ,p (n) ) + |P (e) ,a (n) )) (26) 

I 0,0) = ^(|a( e) ,p (n) ) - |p (e) ,a (n) )) (27) 

| 1,0) = |a w , 0^) (28) 

| 0,0) = |p (e)/ a w ) (29) 



Transitions in an ESR experiment can be induced with a "perturbing" interaction 
of electromagnetic radiation. A resonance condition is achieved when the frequency of 
this incident radiation exactly matches the energy difference between an initial and final 
state (equation 30). The intensity of this transition is the square of the matrix elements of 

fo = SfiMj " ^initial (30) 

the perturbing radiation between the initial and final states (equation 3 1 ). The operator in 

{(initial |H,| final)] (31) 

equation 3 1 , which represents the incident radiation, is defined formally (equation 32) as 
the total dipole moment operator (-^i T ) dotted into the linearly polarized oscillating 
magnetic field (H,). The above equation then transforms into final form as shown in 
equation 33. 

H, = -u T • ff, (32) 

H, = (grf3 e S - gr n P n f) T • H, (33) 

Low field transitions are quite different in terms of their selectivity and intensities 
due to the coupling of states to give the effective quantum numbers: \ F, M F ) . In 
essence there are four transitions with the selection rules of AM F = ±1 by virtue of H, 
being 1 to H ( when H // z and H, // x) where the transition operator becomes equation 
34. A typical evaluation of a low field transition matrix element and corresponding 

H x = gP e H,5 x - g n P n H,l x (34) 

intensity is given in equations 35, 36, and 37 for the transition | l,-l) -* | 1,0) . The 



(P(e)' P(n) \ S x\ a (e)' Pi 



In) 



P(e)' P(n) Kxl P(e)' a (n)) ~ /2 



l,-l) -> | 1,0) intensity a (gp\ cos co - g n P„ sin co) 2 



10 
(35) 

(36) 

(37) 



more common intermediate and high field transitions with H, // x and H, 1 H has a 
transition operator as denoted in equation 38 with a general matrix element given in 
equation 39. Thus, the selection rules for higher fields are in general AM S = ±1 and AM, 
= 0, and the intensity is the square of this element. 



ff, = srp e H,S x 



(38) 



(M s , M T |ff J M s > , M x >) - srp e ff,(M s |S x j M S ')(M X j| M T <) (39) 

A typical experimental arrangement due to technological constraints is to utilize a 
constant frequency source and to sweep a static field. A problem arises in this 
experiment in that the magnetic field is different for subsequent transitions. An excellent 
approximate solution 3 to this situation is given in equation 40. With the values of g and 
v, this equation can be solved for A... It must be noted that if A./g is > then the M, - 
-1/2 line occurs at fields higher than M, = +1/2, and the reverse is true if A./g < 0. 
A 1 



ff = 



SrP, 



V2hv) 



- Mj ± 



M x ~ + 



'-(£) f) "( I + ^ : 



1/2 



(40) 



CHAPTER 2 
THE NATURE OF A SPIN PROBE UNDER THE INFLUENCE OF SUPERCRITICAL 

CARBON DIOXIDE (CO,) 

General Description of Supercritical Fluids 

The behavior of pure liquids and gases are in general fairly well characterized, but 
the conceptual understanding of a supercritical fluid is somewhat esoteric. There has 
been some evidence in the literature 4 "' 4 that supercritical fluids undergo "clustering" or 
local density augmentation particularly near the critical point, which may explain some of 
the unusual macroscopic behavior of these elusive fluids. A substance in the supercritical 
state (especially approaching the critical point) seems to lose any homogenous identity by 
undergoing time and spatial-dependent fluctuations in density. Extreme morphological 
changes in the fluid with little or no change in the temperature or the pressure of the 
system, (i.e.; the singular nature of the isothermal compressibility at this locale on the 
phase diagram), would seem to indicate a struggle on the molecular level for the more 
appealing "microscopic" phase of the moment. 

These extreme molecular environments under the auspices of a single phase 
affords these fluids unique properties that can be intermediate between a gas and a liquid. 
This is evident by the fact that these fluids can possess the solvating power of a 
condensed phase (solvation generally scales logarithmically with density), but on the 
other hand may exhibit the mass transport properties (diffusivity and viscosity) of a gas. 

11 



12 
This unique combination of solvent properties has led many authors" 1516 to describe the 
nature of a supercritical solvent as "tunable." In this case, the "tunability" refers to the 
thermodynamic timescale. But, what is truly happening on a faster timescale? To fully 
realize the potential of supercritical fluids, a fundamental understanding at the molecular 
level must be realized over a wide range of temperatures, pressures, and densities in and 
near the supercritical region. Only after this has occurred will the potential of 
supercritical fluids be revealed. A microscopic feel of these systems seems to be a 
necessity. This is partly due to the extensive commercial interest in supercritical fluids as 
a superlative alternative to traditional halogenated solvents because of their recyclability 
and relatively benign activity towards the environment. 

Introduction 
The recent investigative fervor into the nature of supercritical fluids has created 
some controversy 17 18 about the true behavior of these systems at or near supercritical 
conditions. It is somewhat accepted that pure supercritical fluids possess some degree of 
solvent-solvent clustering. 414 But, what is more speculative is the existence of solute- 
solute association 19 when a dilute solute is introduced into a supercritical system. The 
essence of the question is how does the supercritical solvent treat the impurity? Does the 
supercritical system ignore the presence of the impurity and continue to self-cluster or 
does the solvent fully solvate the solute as it clusters? Is there some degree of solute 
association even in an extremely dilute situation where tiny time and spatial-dependent 
reaction centers are created with the presence of a surrounding modulatory bath of 
supercritical solvent structure? One can envision four or more possible scenarios: 



13 
1) solvent clustering with disregard to the solute impurity (without solute association) 2) 
solvent density augmentation around individual solute molecules that tends to hinder or 
enhance the transport of the solute (which may or may not be dependent on the solvent's 
location on the phase diagram) 3) solute association with minor solvent clustering and 4) 
solute association with solvent density augmentation around a solute cluster. These are 
just some of the questions that we wish to address and to begin to answer in this 
monograph. 

The idea of critical clustering has been bantered about between authors who have 
argued for and against its existence with seemingly varying degrees of conviction. 
Randolph et al. 17 initially reported evidence of critical clustering in and near supercritical 
ethane based on enhanced spin exchange rate constants of a spin probe. On the other 
hand, Batchelor 18 has investigated the spin-rotation line broadening mechanism of very 
dilute (~1 x 10" 5 M) solutions of supercritical hexane and ethanol via a spin probe. 
Batchelor 18 has argued that the spin-rotation mechanism is a more reliable indicator of 
critical clustering rather than spin exchange on the basis that an enhanced spin exchange 
can also be promoted via the onset of gas phase kinetics (and not necessarily critical 
clustering). 

More recently, Randolph et al. 19 have pointed out that without the presence of 
solute-solute association, their results for rotational diffusion models do not produce 
reasonable results. They have argued that previous investigators 5 " 14 have ascribed 
enhanced rotational diffusion times in various supercritical media to solvent-solute 
density augmentation almost exclusively, but they (Randolph et. al.) stress that the 



14 
experimental results they have obtained do not make physical sense unless solute 
association is occurring since the local density enhancement far exceeds liquid densities. 
In this regard, the above authors have conceded previously the likely possibility that 
solute-solute clustering is probably dependent on the Lennard- Jones interaction potentials 
of the particular combination of solvent and solute. 17 

Thus, in order to begin to address the apparently rich phenomena outlined above 
and to try to corroborate or dispute some of the claims of the previous investigators, we 
have undertaken an investigation to explore some of the dynamical properties of a spin 
probe (di-tertiary butyl nitroxide, DTBN) under the influence of supercritical carbon 
dioxide (C0 2 ). Carbon dioxide has been chosen because of its amenable supercritical 
conditions (T c = 31.0 °C, P c = 1070.1 psi and p c = 0.467 g/mL ) 20 and its popularity as an 
exemplary alternative solvent. 

Experimental 

A high-pressure ESR cell fabricated (see Figure 2.1) from 6 mm O.D. and 2 mm 
I.D. quartz capillary tubing traversed a transverse electric (TE102) ESR cavity and was 
mounted horizontally (The ESR cavity was mounted in this fashion to minimize the 
gravitational effect on the concentration near the critical point, 21 and all of the remaining 
components of the system are coplanar to within approximately 1") between the poles of 
an electromagnet. The ESR cell was connected to ancillary components (see Figure 2.2) 
of the system via 1/16" stainless steel tubing. This entire system was enclosed by a 
carefully constructed insulated foamboard enclosure and was subsequently sealed with a 
commercial foam sealant to create a semi-permeable insulating structure that would fully 



15 



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17 
encompass all of the above mentioned components. The temperature of the subsequent 
air bath was regulated and controlled through the operation of two power resistors 
(nominally 110 W) that were suspended within the insulated volume and three 6" 
circulatory fans that were run at a high rate for thorough mixing. The temperature was 
monitored by two calibrated resistance temperature devices (Omega, model 
1PT100K.2010) that were placed within the enclosed environment. The air bath was 
thermostatted to within ± 0.2 C C. The pressure of the apparatus could be varied by a 
piston (High Pressure Equipment, standard model 62-6-10) and monitored with a strain 
gauge pressure transducer (Omega, model PX612 5KGV) to within ± 2 psi. 

To obtain supercritical conditions of carbon dioxide the apparatus was first 
evacuated and flushed several times to ensure that contamination did not occur. The 
vessel was then charged with fresh C0 2 (Scott Specialty Gases. Instrument Grade 
99.99%) up to the maximum pressure of the CO, cylinder and was further pressurized 
with the use of liquid nitrogen to facilitate the condensation of C0 2 . The temperature 
was then brought up to the desired supercritical isotherm and a specific pressure was 
obtained. 

At this point, a high pressure circulating pump (Micropump, model 1805T-415A) 
was initiated to begin the circulation of the C0 2 through the high pressure system. An 
equilibration period of at least two hours ensured that the fluid had reached thermal 
equilibrium. After this allotted period, di-tertiary butyl nitroxide (DTBN, Aldrich, 
90%tech. grade) was introduced into the volume via a high pressure injection valve 
(Valco, model C2-2306) equipped with a 20 uL injection loop. The circulating pump 



18 
ensured mixing of the DTBN and carbon dioxide. The pressure and temperature were 
then noted as the starting conditions, and the concentration of CO, was calculated by the 
appropriate P,V m ,T measurements (based on a total volume of 140 ± 1 mL ) by Wagner 
et.al. 20 The accurate determination of the C0 2 concentration was needed to calculate the 
mole fraction of DTBN relative to C0 2 . The weight fractions of DTBN to C0 2 were well 
within the range of solubility. 22 

After the equilibration period, ESR spectra were recorded with the following 
experimental arrangement: an X-band frequency generator (Varian El 02 Microwave 
Bridge) was coupled to a horizontally mounted TE102 cavity and placed 90° relative to a 
permanent static field (Varian V-3601 Electromagnet). The field was modulated at 100 
kHz with a modulation amplitude of 1 G. The center field was set at 3225 G and swept 
over a 50 G range. After a spectrum was recorded at a particular pressure along an 
isotherm, the mixture of carbon dioxide and DTBN was slowly leaked via a leak valve 
from the system to obtain a new pressure. In this way, a constant mole fraction of CO, 
and DTBN was maintained in order to monitor the behavior of supercritical C0 2 along an 
isotherm over a wide range of pressures (and thus densities) to see the response of the 
fluid with a constant number of probe molecules at each pressure. This procedure would 
give a similar thermal profile for each pressure and density. 

Results and DJaaiSaOfl 
The two dominant line broadening mechanisms that contribute to the overall 
linewidth in these systems will be spin-rotation at dilute concentrations and spin 



19 
exchange at higher concentrations. It has been pointed out 18 that spin-rotation might be a 
better indicator of critical clustering rather than spin exchange because enhanced spin 
exchange near the critical density might just be an indication of the onset of gas phase 
behavior rather than a sign of clustering. A deconvolution of these two line-broadening 
mechanisms is difficult in concentrated samples because of the overwhelming 
contribution from spin-exchange, which tends to mask the residual linewidth (and thus 
the contribution) from the spin-rotation. Therefore, solute-solute and/or solute-solvent 
clustering might best be explored with the observation of spin exchange (v ex ) and 
correlation time (x c ) simultaneously (with the treatment of spin-rotation as a residual 
Gaussian linewidth contribution). In this manner, clustering pertinent to the spin probe 
will be shown more conclusively than by study of either independently. Correlation 
times essentially reflect the extent of time-averaging of the anisotropics caused by the 
modulation activity of the environment surrounding the spin active species. Therefore, 
correlation time will manifest itself in a completely different manner (essentially an m, 
dependence in the ESR spectra) in relation to spin exchange (usually an equal broadening 
of all lines considered). 

To try and circumvent the above mentioned problem, a "modest" amount (~ 1E-5 
mole fraction) of spin label (DTBN) was introduced into supercritical carbon dioxide 
(C0 2 ) to try and account for both the spin exchange (v ex ) and correlation time (t c ). It has 
been reasoned that the dominant mechanism at higher pressures should be mostly spin 
exchange, but as the pressure is decreased towards the critical pressure, spin-rotation 



20 

might have a more significant contribution. Specifically, spin-rotation should be less 
obvious at higher pressures, and more pronounced at lower pressures as there will be a 
marked decrease in viscosity (n) as an isotherm is traversed (spin-rotation has a T/n 
dependence). Therefore, if normal behavior prevails spin-rotation will contribute to an 
overall larger linewidth as pressure decreases. Therefore, the spin exchange frequency 
that will be modeled might be slightly higher (due to the increased contribution of spin- 
rotation at lower pressures) than the absolute frequency, but nonetheless the rate constant 
of spin exchange (k e ) versus correlation time (x c ) should still show the trend sought. 
Otherwise, increased spin exchange would not necessarily point to solute association, 
because it might simply be explained by the onset of gas phase behavior and an increase 
in transport phenomena. Therefore, the prudent representation of the data would be the 3- 
D plot of spin exchange (v e J and correlation time (t c ) versus pressure (or density). 

The general strategy was to prepare a supercritical bath at selected pressures (and 
thus densities) along an isotherm and to extract the motional behavior (i e • v t and 

v j ex' c' ****** 

residual linewidth (lw re J) of the spin probe from ESR spectra. This would be performed 
over several temperatures and mole fractions to determine their dependence or lack 
thereof on the dynamical propeties. 

The ESR spectra were simulated with the use of modified version of Freed's 23 
simulation program that have been tailored specifically for nitroxide spin labels. Freed's 
program explicitly accounts for spin exchange and correlation time and all other 
relaxation processes can be taken into account by a residual linewidth. The simulation of 



21 
these spectra were performed by inputting the g and A tensors of DTBN extracted from 
the powder spectra of Griffiths and Libertini. 24 The spin exchange rate (v e J, diffusional 
rate coefficients (d^, d^, and the residual linewidth (lw res ) were varied to optimize the fit 
of the experimental spectra. The spin exchange rate and diffusional rate coefficients were 
used to calculate the respective rate constants of spin exchange (k e ) and rotational 
correlation times (t c ) appropriate for each fit (see equations 41-43). 

V ex = K [DTBN] 2 (41) 

/ \l/2 

Kyd„) = drot (42) 



6d rot 



Figures 2.3-2.7 show the series of digitized ESR spectra with their respective 
simulations for a series of pressures along the 34 °C isotherm at x = 6.1E-5 (where x is 
defined as mole fraction). Table 2.1 shows a summary of the pertinent thermodynamic 
and motional data for these spectra (Figures 2.3-2.7). Figures 2.8-2.1 1 show a series of 
spectra and simulations again at 34 °C isotherm, but at x - 5.4E-5. Table 2.2 shows a 
summary of these data. Figures 2. 12-2.1 5 show spectra and simulations at 40 °C (x = 5.4 
E-5) and Table 2.3 shows a summary of these spectra. Finally, Figures 2.16-2.18 show 
50 °C spectra (x = 6.4 E-5) and Table 2.4 shows a compilation of these data. 



22 

Figures 2.19-2.24 illustrate the nature of the correlation time and spin exchange 
behavior with respect to density, temperature, and concentration. At all three 
temperatures there is a general increase in correlation time (x c ) as the reduced density is 
traversed from high density to low density, except for the 34°C isotherm at x - 5.4E-5 
(compared to % = 6.1E-5). Here there is a general increase in the correlation time until 
the reduced density approaches approximately 0.68. 

The rate constants (k e ) also increase in general as reduced density is traversed 
(except for the above mentioned example) for the 34 °C and 40 °C isotherms. The 50 °C 
isotherm shows a general increase then a small decrease in rate constants that mirror a 
subtle increase in correlation time. 

Figure 2.24 shows the temperature dependence of the correlation times at a 
constant mole fraction of 5.4 E-5. This comparison suggests that the interaction of the 
solvent becomes more frequent at the lower temperature (34 °C) at an inflection point of 
reduced density at approximately 0.68 and nearly equals the correlation time found at the 
higher temperature (40 °C) at this density. Overall, the correlation time is higher for the 
lower temperature up until this point. 

The temperature dependence of the rate constants (Figure 2.23) at the same mole 
fraction of 5.4 E-5 indicates that the spin exchange rate is lower in general for the lower 
temperature until this reduced density is reached (0.68). At this point, a precipitous 
increase in the rate constant is seen for the 34 °C isotherm that slightly exceeds the rate 
constant observed at the 40 C C isotherm. 



23 
Conclusion 
It appears that the rate constants (k e ) for spin exchange generally increase as the 
reduced density decreases at the lower temperatures (34 °C and 40 °C) nearer the critical 
temperature (T c = 32.1 °C). A precipitous increase in k e is witnessed at the 34 °C 
isotherm and mole fraction of x = 6.1 E-5 at a reduced density of approximately 0.68. A 
concomitant increase in correlation time (x c ) is generally prevalent at these temperatures 
also. This behavior indicates a clustering phenomenon (either solvent-solute or solute- 
solute), otherwise the correlation time should decrease as the rate constant increases if gas 
phase behavior is truly the cause of an increased rate constant. The exception to this 
behavior is the lower mole fraction (x = 5.4 E-5) at 34 °C. At a reduced density 0.68, 
there is a distinct decline in x c with an increase in k e . This would indicate an onset of gas 
phase behavior instead of clustering and points to solute-solute clustering as a possible 
cause of enhanced rate constants in the higher mole fraction (x - 6. 1 E-5) experiment at 
34 °C. 

The behavior at 50 °C indicates that there is a marked increase in rate constants 
and correlation times at a higher reduced density (p R ~ 1 .0) and then a leveling off at 
lower reduced density. This would indicate that clustering behavior might occur at higher 
reduced densities when the temperature is further removed from the critical temperature. 

The related errors in the spin exchange rate constants (kj and correlation times 
(t c ) that are inherent in the simulations can be accounted for with a generous range of 
± 0.5 Lmol's'xlO 13 for k e and ± 5 ps for x c respectively. 



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CHAPTER 3 
VIBRATIONAL SPECTROSCOPY THEORY 

Classical D escription 

The origin of vibrational activity stems from the seemingly simple interaction of 

two mass points that are physically connected through space and are originally at an 

equilibrium position (see Figure 3.1). The impending motion (in the absence of torque or 

any rotational motion) can be described by classical harmonic motion, which is defined as 

the movement that results when a force acting on a body is proportional to the 

displacement of the body from an equilibrium postion. 25 Implementing Newton's First 

Law to describe this motion (see equation 44) results in a homogenous 2 nd order 

differential equation. The general solution to type of differential problem is given in 

d 2 x 
F = ma = m — 7 - kx (44) 

equation 45. If this eigenfunction is differentiated twice, the result is given in equation 

x = A cos (cot + 5) (45) 

46. The angular frequency associated with the linear motion of this system can be related 

2n(-© 2 [A cos (cot + 8) ] ) (46) 

simply to the ratio of the force constant (i.e.; the scalar response to the force acting on the 
body) to the mass (shown in equation 47). The one-dimensional energy of a two-body 

50 



51 




o 

a 



u 

O 

-a 

o 

X> 
I 

o 



to 

i 



52 



1 fk 



V = 



2n \ m 



(47) 



oscillator can be represented as shown in equation 48. 



T = r m i v i 2 + 2 m 2 v 2" + j kx 



(48) 



i?] ■ linear velocity of m, 
v 2 - linear velocity of m : 



1 



dv 



— kx~ s classical potential integrated from F = - 
2 dx 

By using the center-of-mass relationship (i.e.; m,x, = -m 2 x,), differentiating with 
respect to time, and utilizing momenta in lieu of mass and velocity (equations 49 and 50), 



m, 



dt 



— iT?2 



dx 2 
dt 



(49) 



m,v, = -n? 2 v 2 = p, = p 2 (50) 

the energy can be represented in terms of momenta (equation 51). The energy can be 



2i77j 2l77 2 2 



(51) 



eventually reduced to a representation (equations 52 - 54) involving momentum (p) and 



T = - 
2 



2 2 

IUjP + m \P 

m\ni2 



+ - kx' 

2 



(52) 



V = 



m|/n 2 



n7] + m 2 
reduced mass (u), which is commonly known as the classical Hamiltonian. 



(53) 



53 



P 2 1 



T = — + - *x (54) 

Quantum-Mechan ical Treatment 
The classical Hamiltonian for a two-body interaction can be transformed into the 
quantum-mechanical operator via equation 55. The Hamiltonian in essence contains all 

-* 1 2 

— — V> + v(r)v|/ = £ty (55) 

of the energetic information of the system, but the center-of-mass contribution to the 
kinetic energy operator in this instance has been excluded in this equation because it 
represents only a shift in the total energy of the system. When the cartesian coordinate 
system (x. y, z) is reconfigured into spherical polar coordinates (r, 0, <|> ), the Hamiltonian 
converts to equation 56, where J 2 is the square of the angular momentum. The solution 
of this equation involves the selection of an appropriate wavefunction that is separated 



- h 2 
2" 



13 2 dty 

■ 2 dr dr 



1 *2. 



+ T J V + V ^ = Ey (56) 

2|ir" 



into a radial and angular dependence (see equation 57, where Y JM is the spherical 
harmonic). 

\|/ = R(r)Y JiM @,$) (57) 

The one-dimensional radial Schrodinger equation is the result of this substitution 
(equation 58). An effective potential can be written to encompass the potential due to the 



54 



internuclear separation (V(r)) and a "centrifugal" potential 26 of the spherical harmonic 



h 2 d 



, dR (h 2 J(J + 1) 



\ 



2\i dr dr \ 2\ir' 



V ef r= V (r) + V cent 



+ V(r) 



R = ER 



(58) 



(59) 



(equation 59). To simplify equation 58, the substitution of S(r) = R(r)r gives equation 60. 



- h 2 d 2 S 
2ur 2 dr 2 



f*2 



+ 



h L j{j + l) 



> 



+ V(r) 



S = ES 



(60) 



V 2ur z 

A rigorous solution to this equation for the corresponding energy levels and 
wavefunctions requires only a functional form of the potential (V(r)). Generally speaking 
the potential V(r) encompasses the electronic energy E el (r) (obtained from the solution of 
the electronic Schrodinger equation) and the nuclear repulsion term (V^) (see equation 
61). The parametric dependence of the electronic dependence E el on r (and thus the lack 

V (r) = E„ (r) + V NN (61) 

of an analytical form for E el and V^) has led to the empirical development of V(r). A 
popular potential (Dunham potential 26 ) is a Taylor series expansion about r e (equation 
62). Typically V(r e ) is set equal to zero (chosen arbitrarily) and therefore the first 



VCr) = VlrJ + 



dV 
dr 



(r - r ) + 



1 d 2 v 



2 dr' 



(r - r e ) z +. 



(62) 



derivative is zero (equation 63). If the first term is the only one retained (equation 64) in 
the expression and J is set equal to zero, the harmonic oscillator solutions are obtained 
(equation 65), where H v (cc l/2 x) are the Hemite polynomials. The related eigenvalues i 



are 



dV 

dr 



= 



given in equation 66 for the non-rotating harmonic oscillator. 



55 



(63) 



V(r) = - k(r - r e Y 



k = 



d 2 V 



dr' 



(64) 



-ar 



S = NvHv(ay /2 xe 2 



x = r - r„ 



a 



fiCO 



w v = 



2 v v\ 



V7T 



1/2 



£(v) = hv(v +1/2) 



i (kY /2 

v = — — 
2k Vuy 



(65) 



(66) 



Still another possible potential is the Morse potential (equation 67), which 
approaches a dissociation limit [V(r) = D] as r goes to infinity. Furthermore, the Morse 
potential can be solved analytically to give the following eigenvalues with harmonic and 



VCr) = D(l - e" ptr " re) ) 2 
anharmonic terms for the harmonic oscillator (equation 68). 



1 1 7 

G(v) = © e (v + -J - to e x e (v + -r 



(67) 



(68) 



56 



The prediction of allowed vibrational transitions involves the assessment of the 
dipole moment integral (equation 69) in which single primes denote upper levels and 
double primes denote lower states. If the dipole moment (u(r)) is represented as a Taylor 
series expansion (equation 70), the evaluation of this integral is given in equation 71 . It 
is immediately evident that the first term on the right is equal to zero or 1, because the 

MV v" = j (p' Wjb utocp'Vib dr (69) 

du 



H = n e + 



dr 



(r - r e ) 



1 d 2 u 



2 dr' 



(r - rj z +. .. 



MV v" = n e J cp-; ijb (p 



' dr + d ^ 

vib ar + ~~ 

dr 



IV'lib ( r " r e )y\ ib dr+. 



(70) 



(71) 



vibrational wavefunctions are orthogonal. Of the remaining terms, the second term 
(which is the change in the dipole with respect to change in position evaluated at the 
equilibrium position (r e )) contributes the most to the intensity of the transition (see 
equation 72). 



I °c \Mv' v"\ 2 oc 



du 



dr 



(72) 



is 



The remaining component that needs to be evaluated for a vibrational transition 
the integral in equation 73. With the utilization of the harmonic oscillator wavefunctions 
and the recursion formula between Hermite polynomials (equation 74), the resultant 
integral is given in equation 75. The evaluation of this integral leads to the familiar 

J 9' vib Cr - *- e XP"viJb dr (73) 



57 
2xHn(x) = H n+l (x) + 2nH n _,(x) (74) 

vibrational selection rules of Av = ± 1 (fundamental bands only), because the Kronecker 6 
(in equation 75) is v' = v +1 or v - 1. 

1/2, 



(V ' WV) = VW [^ + 15 V,v + l + ^V,V-l] 



(75) 



CHAPTER 4 
INFRARED SPECTRA OF Nb 12 C, Nb 13 C, and Nb0 2 MOLECULES ISOLATED IN 

RARE GAS MATRICES 

Introduction 
The rudimentary architecture of "met-cars" 27 is the metal carbide diatom. Within 
this frame of reference, it is necessary to elucidate the basic interactions between the 
metal and carbon as a duet with and without the perturbing interactions of a solvent 
structure. With this in mind, the gas phase ground electronic state of Nb' 2 C has recently 
been determined to be 2 A V2 by Simard et. al., 28 confirming the original prediction of 
Weltner and Hamrick 29 based on electron spin resonance (ESR) studies of this radical in 
solid rare-gas matrices. In this specific case, it would have proven difficult to detect (via 
ESR) such a species in a rare-gas matrix without the "quenching" of the orbital angular 
momentum. A diatomic radical with an orbitally degenerate ground state (i.e.. n, A, etc.) 
is typically rendered undetectable in the ESR due to the subsequent diffuse nature of the 
signal. But, in some instances angular momentum can be "quenched" by an 
orthorhombic crystal field 1 which causes its properties in the matrix to approach that of a 
Z molecule. 

It is generally the case that rare-gas matrices at cryogenic temperatures provide 
the "mildest" of perturbations on the trapped guests, 3031 indicated by shifts from their gas 
phase vibrational frequencies and electronic levels being only a few percent. 3031 

58 



59 
However, if the molecule is highly ionic (as, for example, LiF) the molecule-matrix 
interaction can be large and the shifts in the solid state considerably larger. If this 
interaction is substantial and the molecule is sitting in an asymmetrical site in the matrix, 
then large anisotropic crystal field effects can also occur which effectively remove the 
axially-symmetric character of the electronic wavefunction, as referred to above. And, 
generally speaking the chemistry in the gas phase versus the condensed or solid phase can 
be radically different as is evident in gas versus condensed phase acidities of some 
mineral acids (e.g., HC1). The entrapment of these molecules in solid rare-gas matrices is 
integral to the complete understanding of the entire continuum of the chemistry between 
the isolated gas phase intrinsic properties and the initimate guest/host interactions of the 
solid and condensed phases. 

The ground state of NbC has been calculated to have an exceptionally large dipole 
moment (p. = 6.06 D) 28 so that even without the quenching of the orbital angular 
momentum one can expect large matrix shifts for corresponding optical spectra when 
investigated in a matrix. Also, through spin-orbit effects there can be mixing of the lower 
states so that forbidden electronic transitions may be observable. With this in mind, we 
now wish to report the effects of the ground state vibrational frequencies (AG l/2 ") of 
Nb 12 C, Nb 13 C, and Nb0 2 when trapped in solid neon, argon, and krypton. 

Experimental 

The experimental setup has described in detail previously. 32 In summary, 
mixtures of niobium and carbon (slightly rich in the metal) were pressed into pellet form 



60 
and vaporized with a highly focused Nd:YAG laser (Spectra Physics DCR-1 1) operating 
at 532 nm. The metal and carbon plume was co-condensed in rare gas matrices at a rate 
of approximately 10 mmol/hr over periods of 1 to 1.5 hour onto a gold-plated copper 
surface at 4 K. The IR spectra were measured via a vacuum FTIR spectrometer (Bruker 
IFS-1 13V) equipped with a liquid-nitrogen-cooled MCT detector (400-4800 cm"') used in 
conjunction with a KBr beam splitter. The spectra were taken with a resolution of 2 cm' 1 
(or in some cases 1 cm" 1 ) and a scan number typically of 200. 

The niobium powder was purchased from Electronic Space Products International 
(99.9% purity). Carbon- 12 was obtained as a spectroscopic grade electrode in graphite 
form and ground into a fine powder. Amorphous carbon- 13 (99% purity) was purchased 
from Isotech and outgassed at 1400 °C for 1 hour before use. The matrix gases were 
obtained from the following vendors; neon (Matheson, 99.9995% purity); argon (Airco 
99.999% purity); and krypton (Praxair, 99.9985% purity). 

Results and Discussion 
Kr 

The most conclusive evidence for the formation of Nb 12 C and Nb l3 C was in a 
krypton matrix shown in Figure 4. 1 (and Table 4.1). Traces A and B show the absorption 
spectra in the stretching regions for the fundamental bands of Nb l2 C (B) and Nb 13 C (A) 
after annealing the spectra to 45 K and quenching to the original temperature of the 
deposition (4 K). By observing the similarities and differences between the two spectra, 
one can discern that the bands to the blue of the sharp band at 941 .0 cm"' in trace B are 



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62 
common to spectra A. Therefore, it is quite reasonable to remark that these absorptions 
are not due to any NbC containing products. The clear differences between the two 
spectra are: 1) the absorption at 941.0 cm' 1 in the lower trace and 2) the absorption at 
908.5 cm"' in the upper panel, which corresponds very reasonably with the isotopic shifts 
that can be calculated by the usual harmonic relationship for diatomics represented in 
equation 76. Both of these absorptions are quite strong in their respective spectra and are 



'13 = v 12 



w 



(76) 



VUl3^ 

absent in comparison with the isotopic counterpart. Therefore, these two absorptions 
have been confidently assigned to Nb l2 C and Nb l3 C respectively. 

The strong peak at approximately 949 cm' 1 has not been assigned, but it appears in 
traces A and B, therefore it is assumed that it is not a Nb (n) C (n) species. It has also been 
assumed that the absorptions (the weak doublet at 970.3 and 967.6 cm" 1 ) that are at higher 
wavenumbers relative to the NbC species are due to the stretching frequency (ies) of NbO 
and/or Nb0 2 . These transient species are prevalent in this type of experiment as has been 
remarked by Simard et. al. 28 and the personal experience of the investigators. 
Unfortunately, the characterization of these species in a krypton matrix has not been 
made to date, so it cannot be concluded for certain that these are NbO (n) impurities. The 
belief that this absorption is indeed due to NbO is based on the fact that these peaks are at 
slightly lower frequencies than those assigned to NbO in neon and argon matrices, 33,37 
which will be discussed further below. The weak absorption at 1024.5 cm' 1 is again not 
believed to be a Nb (n) C (n) because it appears periodically in both Nb 12 C and Nb 13 C spectra. 



63 



Ar 

Figure 4.2 displays the spectra from experiments carried out in a argon matrix. 
Traces A and B show the results of the Nb 13 C and Nb l2 C experiments respectively. The 
stretching frequency due to Nb 12 C is assigned to the band at 952.2 cm"'. The absence of 
this peak in the Nb l3 C spectra (trace A) gives further proof of this assignment. Likewise, 
the absorption at 917.7 cm" 1 that is clearly evident in the Nb' 3 C traces is amiss in the 
Nb l2 C spectra. 

Again, there are residual peaks blue-shifted relative to the above assignments. It 
can be said with certainty in this case that some of these are due to the fundamental 
modes of NbO. Green, Korfmacher, and Gruen 33 have previously assigned these peaks 
(971 cm" 1 , 968 cm' 1 , and 964 cm" 1 ) to NbO in an argon matrix. The three peaks have been 
attributed to NbO in different sites in the argon lattice. Note there is an extra peak at 
974.6 cm" 1 that appears in the matrix containing Nb 12 C. 

In the Nb 12 C trace it is evident that there are some transitions besides those 
attributed to Nb 12 C and NbO that do not appear in the top trace. The doublet that appears 
at 946.2 and 941.4 cm" 1 could possibly be Nb (n) C, but the corresponding 13 C substituted 
frequencies should be present and are apparently absent. Still, another possibility is that 
the Nb l2 C peaks are from molecules in different sites in the matrix. The relatively strong 
peak to the red of the assigned Nb 13 C peak (917.7 cm 1 ) at 91 1 .5 cm" 1 (which may also be 
the peak in the Nb 12 C trace at 913.3 cm 1 ) cannot be definitely assigned. The two sets of 
triplets (913.1, 907.2, 901.1 cm" 1 and 873.8, 867.8, 859.6 cm 1 ) that appear in the Nb 12 C 



64 




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65 

spectrum, but do not appear in this specific Nb 13 C trace (but have been observed in other 
spectra of Nb' 3 C) have been assigned to Nb0 2 , since it has been reported that in certain 
Knudsen cell/mass spectrometric experiments 34 that equal amounts of NbO and NbO, are 
observed to form on vaporization of Nb 2 5 . Further evidence for this conjecture is that 
the triplet pattern (presumably three sites) for both of these transitions mirror the behavior 
of NbO. If this is true, it is the first reported evidence of Nb0 2 in a matrix experiment. 
This will be discussed further. Finally, the peak at 1030 cm' 1 is a transient peak that 
appears in both experiments although it does not appear in trace B for this particular 
experiment. 

It is interesting to note that the gas phase vibrational frequency for NbO (AG 1/2 ") 
has been determined to be 981.3 7 cm' 1 by Gatterer. Junkes. Salpeter. and Rosen. 3536 This 
value is extremely close to the gas phase ground state vibrational frequency of Nb l2 C 
(AG 1/2 " = 980 ± 15 cm"' ) given by Simard et. al. 28 With all else considered, the position of 
the vibrational frequency in various matrices should be a measure of the relative strengths 
of interaction between NbC (and NbO) and the matrix. 
Ne 

The results in neon (Figure 4.3) are the least revealing. In trace B. the weak 
absorption at 983.0 cm" 1 is tentatively assigned to Nb' 2 C. In the upper panel, low 
frequency shoulder of the doublet at 952.8 cm"' and 949.7 cm" 1 is assigned to the Nb 13 C 
stretch, based strictly on the calculated isotopic shift from the band at 983.0 cm" 1 
Furthermore, the gas phase value given by Simard et. al. 28 is very close to the assignment 



66 




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67 
in the neon matrix. This makes intuitive sense since neon is the least perturbing of the 
rare-gas matrices by virtue of the comparison of matrix shifted values for diatomic 
ground state vibrational freqeuncies and electronic transitions. 3031 Again, there are 
impurity peaks due to NbO, which in this case have been previously characterized by 
Brom, Durham, and Weltner. 37 They have assigned three absorptions due to sites in the 
matrix for NbO at 965.5, 973.0, and 977. 1 cm"'. In our spectra, it seems that the only site 
we see is the one at 973.6 cm" 1 . This would not seem to be out of the realm of possibility 
since the matrices were prepared in dissimilar manners. 37 The peak at 989.4 cm' 1 that 
appears in both spectra has not been assigned and probably is not due to NbO since these 
peaks have been characterized previously, but it seems to be an impurity that cannot be 
related to any Nb (n) C (n) product either. The transition at 914.5 cm' 1 could possibly be due 
to Nb0 2 since only one site is seen here for NbO (in contrast to the argon data). 
The Ground State ( 2 A^) Vibrational Properties of Nb l2 C and Nh l3 C 
Table 4.2 gives a summary of the ground state harmonic frequencies (co e ) and their 
corresponding anharmonic corrections (co e xJ that were calculated from the 
experimentally observed AG" 1/2 frequencies for the two isotopomers. The relationships 
used in these calculations are given in equations 77 - 80. 38 A comparison of the matrix 
and gas phase values are given along with harmonic force constants. 

AG'i /2 (Nb 12 C) = v l m 2 = col 2 ~ 2toi 2 xl 2 (77) 

®i 3 = pco] 2 (78) 

°>lh? = pVxi 2 (79) 



68 

p = (u 12 / u 13 ) 1/2 = 0.9652 (80) 

The NbCXmolecule 

The fortuitous observation of Nb0 2 in an argon matrix has given us a chance to 
speculate on the ground state of the Nb0 2 molecule. The ground electronic state of Nb0 2 
has not been reported to our knowledge, but we would like to report the ground state 
vibrational frequencies of v, and v 3 , and cautiously assign the v, bending mode. The 
symmetric stretch (v,) has three sites at 913.1, 907.2. and 901.1 cm" 1 (see Figure 4.2). The 
asymmetric stretch (v 3 ) has three sites as well at 873.4. 867.8, and 859.6 cm' 1 . The 
bending mode would be expected to have a weak transition to the red of the above 
absorptions. We have tentatively assigned the triplet around 518 cm' 1 (524.5, 517.7, and 
510.6) to v 2 (see Figure 4.4). 

With the apparent observation of a symmetric and asymmetric stretch, a bent 
ground state seems to be in order. The bond angle of this ground state can be estimated 
via the ratio of the relative intensities of v 3 and v, with the assumptions made by Ozin et. 
al. 39 Figure 4.5 shows the enhanced region of the Nb0 2 stretching region of Figure 4.2. 



— I = tan 2 

IV, 



( M Nb + 2M sin 2 N 
KM m + 2M cos 2 0y 



(81) 



The ratio of v 3 and v, is based on the integrated intensities of the respective matrix sites. 
The intensity ratio was determined to be 1.71, which gives a bond angle of 20 = 103°. 
This value agrees quite reasonably with Spiridonov et. al. (0 = 101.6(3.3)). 40 

Surprisingly, a theoretical treatment of NbO : has not been cited in the literature 
tothe best of our knowledge. The most recent experimental work on NbO, was an 



69 



Table 4.1. Observed IR bands (cm 1 ) of niobium monocarbide and niobium dioxide in 
rare gas matrices. 



Molecule 



Neon 



Argon 



Krypton 



Assignment 



Nb 12 C 


983. 


952. 2 


Nb 13 C 


949. 7 


917., 


Nb0 2 




873. 8 , 867. 8 , 
859. 6 


NbO : 




913.,, 907.,. 
901.,, 


NbO, 




524. 5 , 517,, 
510. 6 



941 



•o 



908., 



AG' 



AG' 



12 



i : 



v, (sym. str.) 
v, (asym. str.) 
v 2 (bending) 



Table 4.2. Vibrational frequencies (cm 1 ) of the ground state ( 2 A 3/2 ) of Nb l2 C and Nb 13 C 
in rare-gas matrices at 4 K. 



Species 



Nb 12 C 



Parameter 



Ne 



AG' 



i : 



(0 



12 



983. 



1013* 



Ar 



952., 



962. 



Kr 



941. 



951. 



12 12 


15. 


4. 9 


5.o 


Nb l3 C AG",, 


949., 


917., 


908. 5 


co^ 3 


975.2 


926.1 


915.6 


»i 3 xi 3 


12.8 


3.4 


3.5 


k b (mdyn/A) 


6.42 


5.79 


5.66 


'from reference 28. 









Gas Phase 3 



980 ± 15 






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72 
electron diffraction study performed by Spiridonov et al. 40 They reported based on the 
refinement of their data with v 3 fixed at 1009 cm" 1 that v, and v : were 854(41) and 
527(41) cm' 1 respectively. The values reported here are well within the error limits given 
in the above study. Admittedly, they indicate that their treatment is very sensitive to 
temperature, thus the reported vibrational frequencies have a parametric dependence on 
temperature with a generous range of error 2678(39) K. In any event, the vibrational 
frequencies determined in this experiment should give a clearer picture of the true 
frequencies regardless of subtle matrix effects. 

Conclusion 
The experiment performed in krypton gives the best evidence that Nb l2 C and 
Nb 13 C were indeed synthesized. The isotopic shift in frequency of Nb l3 C (AG", /2 = 908. 5 
cm 1 ) relative to Nb 12 C (AG" 1/2 « 941. cm 1 ) is quite satisfactory in comparison to a 
calculated harmonic prediction (AG" 1/2 = 908. 2 cm 1 ). The krypton spectra (Figure 4.1) 
clearly shows these isotopomers with very little interference from impurities. The 
frequency of Nb 12 C in the krypton matrix (AG" l/2 = 941. cm 1 ) has been substantially 
reduced as compared to the gas phase value reported by Simard 28 (AG", /2 = 980 ±15 cm 1 ). 
The percent shift relative to the gas phase value is 3.9%. This is not atypical, as trapped 
species in more polarizable matrices tend to have higher shifts relative to the gas phase. 30 
A comparison of other highly ionic species that have been trapped in matrices shows that, 
for instance, LiF (dipole moment, u - 6.33 D) has a shift in a krypton matrix of 6.9% 
relative to its gas phase value. 3031 This would further validate the approximate magnitude 
of the dipole moment that was calculated by Simard 28 (u = 6.06 D). 



73 

The argon spectra (Figure 4.2) seem to exhibit bands of Nb l2 C and Nb l3 C. but are 
complicated by impurity peaks essentially due to NbO and Nb0 2 . The isotopic shift in 
frequency of Nb l3 C (AG", /2 - 917. 7 cm' 1 ) relative to Nb ,2 C (AG" 1/2 - 952., cm' 1 ) is again 
quite good compared to the calculated harmonic prediction (AG",, 2 = 919. cm" 1 ). The 
percent shift of Nb l2 C relative to the gas phase value is 2.8%. 

The neon spectra (Figure 4.3) was the most speculative in terms of definitely 
assigning bands to Nb l2 C and Nb 13 C, which was discussed earlier. The bands that were 
assigned to Nb l3 C (AG",, = 949. 7 cm' 1 ) and Nb l2 C (AG" l/2 - 982.0cm' 1 ) agrees well with 
the isotopic frequency compared to the calculated harmonic value (AG" I;2 = 947. 8 cm' 1 ). 
The percent shift of Nb 12 C relative to the gas phase value is -0.2%. This percentage is not 
surprising since neon is the least perturbing matrix of the rare gases and most closely 
resembles gas phase conditions. 30 



CHAPTER 5 
DENSIMETER 

Introduction 

A complete understanding of the world that we "see" relies on the ability to 
measure the behavior of a single atom as an isolated entity, as well as the collective 
behavior of various ensembles of atoms and molecules that are placed under varied 
environmental scenarios. Ongoing technological advances have enabled scientists to 
actively research these extreme disciplines. For example, the ability to detect and 
manipulate a single atom/molecule has recently been reported by several groups. 4142 At 
the other extreme, condensed phase physicists have recently discovered the existence of 
the Bose-Einstein condensate, predicted by Einstein nearly 70 years ago. 43 

The unusual behavior of a bulk state is not necessarily confined to the exotic 
conditions of a Bose-Einstein condensate, but can also be realized in conditions of normal 
fluids and mixtures that are far removed from the ambient. This behavior is particularly 
evident near the critical point (P c , p c , T c ), due to time-dependent fluctuations in density. 44 
As a result, these regions of a phase diagram can be difficult to define thermodynamically 
due to deficiencies in thermophysical measurements. It has been pointed out by 
Wagner 20 that "the IUPAC Thermodynamic Tables Project Centre at Imperial College 
London is committed to revise its tables on carbon dioxide, 45 because the equation of 

74 



75 
state used for establishing those tables show quite large systematic deviations from the 
experimental (p, p, T) and caloric values, especially along the coexistence curve and in 
the critical region." 20 Thus, the need for reliable equations of state within the 
experimental uncertainty of physical measurements is needed. 

Furthermore, with the peaking interest of supercritical mixtures as a media for 
novel chemical activity, the ability to predict the behavior of supercritical mixtures from 
the complete and accurate thermodynamic surface of the media of interest is tantamount 
to the complete exploitation of this area. These sentiments have been elucidated in the 
literature by Poliakoff: 46 "... reaction mixtures have to be homogeneous in order to exploit 
the advantages of supercritical fluids. Therefore, a knowledge of their phase envelopes 
and critical points is crucial. Even the simplest reaction system will be a ternary mixture 
(reaction, product, and solvent)... and traditional methods of phase measurements (view- 
cell or sampling) are often very difficult to apply to some ternary mixtures where the 
density differences are small." 46 

The reoccurring theme of these observations, as stated in the literature, is the 
ability to measure the state variables (i.e., p, p, and T) very accurately and with the 
utmost precision. The state-of-the-art techniques for these measurements are adequate for 
temperature and pressure, but density determinations are non-trivial, especially if a 
system is well removed from the ambient. But, why is density so important? 

Thermodvnamir Relqtinnf?hip<f 
The ability to control the behavior of bulk chemical systems involves using the 
predictive power of a thermodynamic equation of state that is developed based on the 



76 
physical property measurements of these aforementioned independent variables. If these 
variables are satisfactorily measured, all of the other thermodynamic properties of a fluid 
can be expressed in terms of the Gibbs energy G (T,P) or in terms of the Helmholtz 
energy A (T,p). It has been pointed out previously 47 that for regions of low or moderate 
pressures, either set of variables is fully satisfactory. However, when the two-phase, 
vapor-liquid region is involved, the Helmholtz energy (A) is much more satisfactory 
since it is a single valued function of T and p. This results from the density (p) being a 
multi-valued function of temperature and pressure in this region of the phase diagram. It 
is possible to tabulate or write separate equations for vapor and liquid properties in terms 
of the Gibbs energy (G) (as functions of temperature and pressure) but, if one wishes a 
single comprehensive equation of state, one uses T and p as variables and the Helmholtz 
energy (A) as the parent function. 

The parent function in both cases is the sum of two functions, one for the ideal- 
gas or standard-state properties and the other for the departure of the fluid from ideal 
behavior. Equations 8 1 and 82 show the Helmholtz energy (A) developed in terms of an 
ideal gas and its relationship to its standard state properties, where p° = P°/RT (P° = 1 bar) 
and G° = A + RT. The departure of the Helmholtz energy from the ideal case is related 
by the following equations (83 and 84), where the compression factor z is defined as V m = 
zRT/P' and P' = p'RT. 

A id (T, p) - A°(D = RT In ip / p°) (81) 

= e°en + rti-i + in (pan i (82) 



77 



[d (A - A^) I dp] = V RT - RT I p (83) 



p 

A(T, p) - A 1 (T, p) = RTl (z - \)d In p' (84) 



As an aside, it should be mentioned that the development of an equation of state 

is often given for the Helmholtz energy (A) instead of a compression factor (z), where 

one first divides A into an ideal and a residual term (Equation 85). The compression 

A = A id + A res (85) 

factor is then defined in terms of a residual Helmholtz energy (A res ). But, if the 
compression factor is initially defined as was the case here, the residual Helmholtz 
equation (A res ) is defined in equation 86, which essentially represents the departure of the 
Helmholtz energy (A) from the ideal case (A id ). The final equation for the Helmholtz 
energy is therefore equation 87. Substituting an appropriate equation of state for the 

P 

res f 

A / RT = J (z - l)d In p 1 (86) 



ACT, p) = G°(T) + RT[-\ + ln(pRT)] + A res (87) 

compression factor (z-1) developed in terms of density, (the virial equation for example) 
gives the complete Helmholtz energy (88), which can be utilized to derive most of the 
other thermodynamic relationships. 



78 

ACT, p) = G°(T) + i?T[-l + In(pRT) + Bp + Cp : +. . . ] (88) 

It is obvious that a nearly complete thermodynamic description of a normal fluid 
can be obtained with the above relationships, but they are contingent on the accurate and 
precise measurement of the state variables with density as the most elusive. With these 
points in mind, the goal at the outset was the development of an all encompassing density 
measuring instrument with the following criteria: 1) accuracy /preciseness, 2) sensitivity, 
3) robustness, and 4) self-calibrating abilities (as an enhanced feature). Thus, the focus of 
this study was the development of a state-of-the-art Archimedes type densimeter that in 
principle would measure fluid densities over a wide range of temperatures and pressures. 

First Principles 
An Archimedes type densimeter relies on principles developed around 200 B.C. 48 
by one of the greatest scientists of antiquity, Archimedes. These principles are based on 
the buoyancy force that is subjected to objects submerged in a fluid. For instance, the 
forces that act on a rigid body that is immersed in a static, isothermal fluid are: 1) 
Newton's second law of motion and 2) the buoyancy force (see Figure 5.1). The former 
is the gravitational force (equation 89), where F, is Newton's force of gravity, m is the 
mass of the object, g is the acceleration of gravity, and h is a unit vector directed along 
the center of the masses of the two bodies. The latter force (p ) (equation 90) is 

F = mgn (89) 

dependent on the density (p) of the medium and the volume (V) of the fluid displaced by 



79 




CO 

I 

3 



80 
the object. The local acceleration of gravity (g) and n are defined as they were in the 
previous equation. These two forces act in opposition, and thus the resultant force 

F = pVgn (90) 

B 

experienced by the immersed object is the intrinsic weight ( F ) less the buoyancy 

Q 

F = (mg - pVg)ii (91) 

res 

force (F ) (see equation 91). It is now evident that in principle it is possible to solve for 

B 

the density of an unknown fluid if the force of a submerged mass could be measured. 

Design of Densimeter 
The ability to measure the density of an object under ambient conditions is a 
relatively simple matter. But, the sampling of density far removed from ambient 
conditions is a non-trivial measurement. If one recognizes from equation 91 that the 
density can be ascertained from the knowledge of the net force applied to an object in a 
fluid, a density measurement would ostensibly be feasible. But, the net force on an 
enclosed object under extreme pressures and temperatures is a very difficult 
measurement. To begin to mitigate this problem, a spring can be used to suspend the 
body within the fluid. Consequently, the force applied by the rigid body can be related to 
the spring's displaced position. If the spring obeys Hookes' Law (which is an excellent 
approximation over a very short deflection range), the deflection will be linearly 



81 

dependent on the total or net force placed upon the system (equation 92). In this 
expression, k is the force constant of the spring and x is the displacement of the spring 
from its equilibrium position. 

F = -kxn (92) 

If we consider the suspension of a sphere from a spring that has been described 
above, the net force experienced by an individual sphere is now incident on the spring 
and causes an expansion that is characteristic of the resultant load (equation 93). In this 

F = (mg - pVg)n = -kxn (93) 



nat 



experiment, two rigid spheres (with masses m, and m 2 and respective volumes V, and V 2 ) 
were placed in a rack and suspended from a spring in a static, isothermal fluid (see Figure 
5.2). With this arrangement, it is possible to measure the force (specifically the position) 
of four separate loads. Four linear equations are the result of the above measurement of 
position as a function of the fluid's density (see equations 94 - 97). The displaced 

* = (m o g - pvgfi = -kx Q n (94) 

F = (m : g - pVg + m Q g - pVg)n = -kx^ (95) 

F = (m 2 g - pV g + m o g - pV g)n = -kx 2 ii (96) 

F = (m 3 g - pV g + m Q g - pVg)n = -kx 3 H (97) 

position of the spring due to the net force of each loading was monitored by a 

commercially available linear variable differential transformer (LVDT), which transduced 



82 




s 

o 

V) 



(N 

i 

00 



83 

the displacement to a voltage. The resultant equations that were used to solve for density 
are represented below (equations 98 - 100), where S is the signal in terms of voltage and 
K is the linear conversion of effective mass to voltage (At the outset, this constant is 
assumed completely linear over the deflection range of interest.). The fourth 
measurement (see equations 94-97) is a linear combination of the other three, thus there 

F = (m q - pV q)ii = -kx n = S K(x) (98) 

r v ' 

F = (m g - pVg + m o g - pV o g)ii = -kx a = S K(x) (99) 

F = (m 2 g - pVg + m o g - pVg)n = -kx a n = S 2 K(x) (100) 

are only three resultant equations with three unknowns ((m -pV ), Kg/k, and p), which 
can be solved to ascertain density. Note that the scaling constants (Kg/k) in these 
equations are grouped together as one term. 

The elegance of this particular design allows for the determination of the density 
without the explicit knowledge of the local acceleration of gravity (g), the force constant 
of the spring (k), or the LVDT constant (K). For example, the spring constant of the 
material will vary with temperature and pressure, but it will not vary over a single density 
measurement. Secondly, the local acceleration of gravity (g) should vary negligibly over 
the distances (0.100") in this experiment. And, it assumed that a linear response will be 
provided by the LVDT. Therefore, these consta nts cancel out of the resultant system of 
equations when thev are solved. And, in theory, the instrument is a self-calihratin ? 
device, 



84 

Densimeter Development 

To bring to experimental fruition the highly attractive theoretical features of this 
type of densimeter, an arduous trial and error procedure ensued. The critical component 
and ultimately the foundation of this instrument was the sensing device that measured the 
uniaxial displacement of the spring with a resultant load. As was mentioned in the 
previous section, an extremely short deflection range was needed to ensure the linearity 
of the deflected spring. Thus a sensing device was needed that had the combination of a 
reasonably short stroke coupled with a high resolution of linear displacement. Therefore, 
an LVDT (Linear Variable Differential Transformer) sensing device (Trans-Tek, Inc. 
model #240-0015) was purchased with a deflection range of 0.100" and a resolution of 
± 1 micron. The constituent parts of an LVDT consist of a weakly magnetic core (0.99" 
OD X 0.492") and a cylindrical transformer with a cored center. The position of the 
magnetic core relative to the transformer of the LVDT gives a linear voltage response 
when it is displaced from a null position (a voltage of zero) to ± 6 volts 
corresponding to ± 0.50" displaced distance. 

With the desired specifications of the LVDT in hand, the next most important 
aspect of the design and engineering process was to find a spring material that would lend 
itself to the rigors of this experiment and more importantly have an extremely elastic 
behavior over wide ranges of pressures and temperatures. In addition, the spring would 
have a deflection of approximately 0.080" with a given load to take advantage of the 
maximum stroke of the LVDT. The spring material that was chosen for use in this 
apparatus was fused quartz. Quartz was chosen because of its extremely elastic behavior 



85 

under environmental stresses (a = 5.5e-7 cm cm' 1 °C''), 49 and its resistance to hysteresis 
when a load is placed on it over time (due to its amorphous structure). A stock piece of 
fused quartz (0.06" OD) was heated, wound around a mandrel and placed in a lathe to 
generate a coiled spring with the approximate dimensions of 0.25" OD and 0.5" Length. 
Excellent control of the spring constant could be exercised with bench top measurements 
of deflection by etching the quartz (2.5 M) over carefully monitored time periods to 
obtain the desired deflection of approximately 0.80" with a load of 25g. 

From a previous design of another densimeter model." two hollow stainless steel 
spheres were fabricated and used as a changeable load that was to be placed on a rack. In 
the current design, the rack and the spheres would then be suspended from a spring and 
the displacement sensed by the LVDT. The sinkers were originally designed to have 
approximately the same volume (V,=V 2 ), but one sinker would have twice the mass 
(2m,=m 2 ). They were machined from hemispheres made of 304 stainless steel with a 
0.625"OD and a 0.437" ID, which gave a sufficient wall thickness to prevent any 
dimensional change (and therefore volume change) up to 3000 psia. A copper bead that 
had the mass of two hemispheres was placed in one set of hemispheres (to achieve 
approximately twice the mass of the other), and the two sets of hemispheres were then 
Heli-Arc welded at the seam. The masses and volumes of the spheres were determined 
via an analytical balance and pycnometric volume determinations. 50 

The remaining parts of the densimeter were built around the specifications of 
these integral components. To begin with, a rack to hold the buoys was fabricated to 



86 
suspend from the quartz spring. The rack was designed to incorporate the LVDT 
magnetic core into a suspension post that screwed into the rack so as to make a single unit 
(see Figure 5.3 for schematic). This rack was assembled by brazing two 0.062" diameter 
stainless steel rods onto a brass disc. The actual supports for the spheres were short 
pieces of stainless steel wire that were brazed in a V-shape to these rods. The brass disc 
was drilled an tapped 5-40. Two brass posts were then fabricated to connect the brass 
disc to the quartz spring, and linked between them was the magnetic core of the LVDT. 
The first rod was threaded 5-40 on one end to place into the brass disc, and the other end 
was threaded 1-72 to place into the LVDT magnetic core. The second brass rod had a 
small hole drilled in the center of a small flattened section to accommodate a hook that 
extended from the quartz spring to link the rack to the spring. And, finally the other end 
of this second rod was threaded 1-72 on one end (to place into the other end of the 
magnetic core). 

With the rack unit in place, there was the need to independently lift the buoys 
from the rack. Mechanical lifters were designed to fit into a cylindrical, brass sleeve 
(1.049" OD), which in turn would slip fit around the rack and spheres. Vertical slides 
were dimensioned to provide proper clearance for each individual loading so as to 
prevent any mechanical contact between the rack and the sleeve and to ensure that the 
load would hang freely from the spring. The lifters were actuated via a magnetically 
coupled cam system that lifted the spheres from the rack to provide the instrument with 
four independent loads. 



87 



C/l 

o 
B. 

V) 

to 

u 
X> 

o 



c 

U 
u 

u 
X) 

-o 

a 

2 

3 
O 



o 
u 

f- 
Q 
> 

u 

■B 



E 

u 

o 

C/J 






CO 



88 
The aforementioned brass sleeve (Figures 5.4 - 5.9) was machined with two slots 
that were 90° apart to accommodate the lifter slides. These slides were made from brass 
(3.990" x 0.185" x 0.145") and were fitted with stainless steel forks (1/32" diameter) that 
were brazed into the body of the lifter slides. The forks had three points that provided 
support for spheres when lifted from the rack. Finally, the lifter slides were slotted at the 
bottom and fitted with small brass wheels to ride on the cam assembly to actuate the 
motion of the lifter slides (see Figure 5.10). The magnetically driven cam assembly 
(Figure 5.1 1) is a two-tier cam equipped with an internal magnet, which is coupled to an 
external magnet. The cam has a 45° step with a total rise of 0.250". The quadrants of the 
cam allow for the motion of the lifters to give four separate loads referred to earlier. The 
cam is fitted onto a magnetic turner (Figure 5.12) by a stem that locked into position by 
a set screw. The cam rides on brass ball bearings (Figure 5.13) placed in between the 
cam and the magnetic holder. In turn, the cam and magnetic turner unit rotates on a set of 
brass ball bearings that sets on the bottom of the densimeter can. A ferromagnet was 
placed into a drilled out section of the magnetic turner, which was held in place with a set 
screw. The ferromagnet was coupled to an external magnet that drove the cam assembly. 
At this point, the densimeter was assembled into two main components (see 
Figure 5.14 for schematic assembly drawing). The upper component consisted of the 
following pieces: 1) a top flange (Figure 5.15) equipped with a brazed spring mount, a 
ball bearing support clamp for a drive screw (see Figure 5.16), a drive screw (Figure 
5.17) that actuated the movement of the LVDT housing (Figure 5.18 and 5.19), and a top 



89 

flange cover (Figure 5.20). 2) a bottom flange (Figure 5.21) that was married to a 
complimentary flange on the densimeter body and supported the top flange via support 
legs (Figure 5.22). The top and bottom flanges were physically joined by an inconel™ 
sleeve (see Figure 5.23) that traversed the LVDT housing and encompassed the LVDT 
core. 

The lower component was made up of a densimeter body (Figure 5.24) that 
contained an end cap (Figure 5.25) brazed into the bottom, a magnetic turning unit, and a 
cam assembly. A flange (shown in Figure 5.24) was brazed onto the densimeter body 
(complimentary to the bottom flange) in order to marry the two components together and 
create a high pressure seal. This seal utilized a copper gasket that was fabricated by 
cutting a piece of 12 gage copper wire to a length of 3.92", and the ends Heli-Arc welded 
together to make a continuous piece. The copper gasket was implemented by initially 
placing it around the diameter of the densimeter body and resting it on the offset of the 
bottom flange. The upper was then placed on top of the bottom flange and the twelve 
10-32 bolts were tightened to compress the copper wire into the gland between the two 
flanges. The gasket is dimensioned to fill 95% to 98% of the triangular cross-sectional 
area of the gland. 

Experimental 

To initially test the viability of this instrument; temperature, pressure, and density 
data for argon and carbon dioxide were acquired over several isotherms near ambient 
temperature, but over a wide range of pressures and thus density. These tests would bring 



90 







»At #*• Mm 



Figure 5.4. The cut-away view of the brass sleeve. 



91 



12:00 and 6:00 position 



H O.tIM i— 



l» Ml— Vtru 

















t.oto 
















2.1 


00 



















Figure 5.5. The 12:00 and 6:00 position of the brass sleeve. 



92 






CO 

o 

o 

o 




o^ 



> 
u 

In 

3 



u 
o 

■si 

c 
o 



si 

o 

> 

■ — I 

<L> 

Q. 



00 



93 



1:30 position 




VUa • .M7 i—r 



Figure 5.7. The 1:30 position of the brass sleeve. 



94 



o 




95 



4:30 position 




.tit widm i 



l§§ wUm <A» 



ItS aiii • .147 tarf 



Figure 5.9. The 4:30 position of the brass sleeve. 



96 



/ 
/ 

/ 
/ 

/ 
/ 

/ 
/ 

/ 
/ 



(J 

cn 




■ 

o 
c 
o 

o 
E 
u 

X) 

T3 

V 
*— 
«3 

3 

s 

U 



u 

u 

— 
o 

1 

< 

o 



op 



97 



cam 



1.030 dia 



.1X6 tfcru on 

s/ie r 




MATL. 

SS 



*9.T4F —J 0.3/5 



0.115 




.1 "" I" 



<-*0 tep thru 



Figure 5.11. The cut-away and top view of the cam. 



98 



magnet holder 



i h 



0.125 



.125 drill thru 
on 5/18 R 




Jlat for nt screw 

grwut top and bottom 
.125 vridt X .020 dttp 

10-32 tap thru 



MATL. BRASS 



Figure 5.12. The cut-away and top view of the magnet holder. 



99 



shims 





•-tort «/K x HI ter 



(»**«!/» fa) 




AM 7X. 
BRASS 



Figure 5.13. The cut-away and top view of the brass shims. 



100 



Quartz Spring 



Drive Screw — 




Flange Cover 



Top Flange 



LVDT Housing 



Bottom Flange 



Suspension Rack 



Densimeter Body 



Cam Assembly 



Internal Magnetic Turner Assembly 



End Cap 



Figure 5.14. The assembly drawing of the densimeter. 



101 



top flange 



lit nut . </M < 




-I"- I- 









t««m 4 **• 



*■""! Ura ••* lap <-U 
• •!<••» 1 W» «te U» 




mail, ss 



Figure 5.15. The cut-away and top view of the top flange. 



102 



support clamp 



0.807 




#33 driii thru 
2 pics 



matl. brass 

3/32 thk 



Figure 5.16. The support clamp that is mounted on the top flange. 



103 



drive screw 




-i h*- 



spacer 



— ' *.*n — 
i 





*.M> j- M 



spacer 



-l/M to M /U •< —Mi 

—4 • Jff t— to iml l i | I I u 



-i ••»• r- 



0Z0z: 



AWTZ,. 5*5* 



Figure 5.17. The drive screw assembly. 



104 



t** 4H* X 600 4mp 

— fc-nt JH X . JM 4m» 



coiZ holder 




MATL. BRASS 



Figure 5.18. The cut-away and side view of the coil holder. 



105 



stop pin 




MATL. SS 



Figure 5.19. The stop pin that is mounted in the LVDT housing. 



106 



cover flange 
top 




side 



»/*M flu Mm 




MATL. SS 



Figure 5.20. The cut-away and top view of the top flange cover. 



107 



BOTTOM FLANGE 




>/M 4*a >n 



AWT 1 /,. SS 



Figure 5.21. The cut-away and top view of the bottom flange. 



108 



support legs 



drill ♦ lap 10-32 
1/2 d—f b*th twtdt 



/ 




ivrwncK Jlats 



—| O.SIt [— 



MATL. 

SS 



Figure 5.22. The support legs mounted between the top and bottom flanges. 



109 



magnet sleeve 



csT- 



nam .136 thru 



2.62£ 



U — 

— \ \— 0.158 



mail, inconel 



Figure 5.23. The inconel magnet sleeve brazed between the flanges. 



110 



body 



a 




MATL. SS 



Figure 5.24. The densimeter body. 




Ill 



a, 







CO 

o 

CO 



o 



112 
to bear the overall operational ability of this prototype instrument and allow for any 
design modifications that would need to be undertaken. 

As mentioned previously, in order to obtain nearly complete bulk thermodynamic 
information, density along with temperature and pressure need to be sampled 
concurrently. In these experiments, the densimeter was coupled with state-of-the-art 
pressure and temperature measuring devices. A Sensotec model TJE/743-03 strain gage 
pressure transducer was used in conjunction with a Beckman model 610 electronic 
readout to display and sample the pressure data. The Sensotec transducer was calibrated 
with a Ruska (model 2465) standardized dead weight pressure gage. The temperature 
was monitored with a four wire platinum RTD (resistance temperature device) interfaced 
to a Keithley digital multimeter (model K-196). This RTD was calibrated versus a 
standard RTD (HY-CAL engineering, N.I.S.T. traceable). The block diagram in Figure 
5.26 shows the schematic setup for the densimeter and the ancillary components. The 
densimeter, pressure transducer, and the RTD are enclosed in an insulated temperature 
controlled oven equipped with six circulating fans that was retrofitted with a newly 
developed liquid nitrogen refrigeration control valve (see reference 50 for details). A PID 
control algorithm monitored the valve and provided temperature control to ± 0.01 
mK. All of the above mentioned devices were interfaced to a personal computer (486 
internal processor operating at 33 MHz with 16 megabytes of RAM) via the standardized 
IEEE general purpose interface bus (GPIB) (see Figure 5.26). 

The actual data collection procedure followed the following protocol: 1) The 
densimeter along with the ancillary components (RTD, pressure transducer and the 



113 




D 



1/5 

c 

U 
T3 

<U 



e/J 
O 

'2 
2 

y 
o 



1) 

J3 



2 
'a 

M 
o 

jo 

H 

^> 

<N 

<0 






114 
accompanying connective hardware) that were contained within the insulated oven were 
first charged to a approximately 1 00 psia with the gas of interest. A vacuum was then 
applied to remove the gas and any lingering gaseous contaminants, and this procedure 
was repeated several times to ensure that impurities were driven from the system. 2) 
After the system was charged to the desired pressure, the oven was started. The oven 
generated heat from the operation of the fans that was countered by the actuation of a PID 
controlled liquid nitrogen valve (previously mentioned) interfaced to the personal 
computer. A desired temperature setpoint was entered into the PID algorithm and 
equilibration of the desired temperature (±10 mK) was obtained within 2-3 hours. 3) 
After the equilibration time, the magnetically driven cam assembly of the densimeter 
was manually turned to a position that corresponded to a specific load. The displacement 
of the quartz spring due to the effective load (corresponding to the competing forces due 
to mass and density) was linearly transduced to a voltage by the LVDT and its circuitry. 
The voltage was acquired with an Iotech model ADC488/8SA analog to digital converter 
(ADC) that sampled the voltage at a rate of 1kHz. The final voltage then for this 
particular reading was an average of 16,384 voltage readings averaged over 16.4 seconds. 
During this acquisition, the temperature was acquired by sampling the resistance of the 
RTD from the Keithley model K-196 digital multimeter. The pressure was sampled 
simultaneously from a Beckman model 610 electronic readout used in conjunction with 
the Sensotec model TJE/743-03 strain gage pressure transducer. The cam assembly was 
turned to a new position (a new effective load) and the above procedure was repeated 



115 
until all four loads were sampled. The acquisition of four loads consisted of one trial. A 
minimum often trials were obtained for this particular pressure, temperature, and density. 
Several runs were obtained along the same isotherm, corresponding to different 
densities. This was accomplished by bleeding the system of some fluid to obtain a new 
density. Thus, the P.T, p surface was obtained by reloading the system with a suitable 
density and sampling at different isotherms. 

Results 

Arson 

Argon (Bitec. 99.99% purity) was chosen to initially test the viability of the 
densimeter, because of its nearly ideal behavior over the temperature and pressure regions 
selected for the testing. The pressure range for the tests were from approximately 550 
psia to 1750 psia at temperatues of 278.1 5K, 288. 15K, and 298. 15K. This corresponded 
to a density range of approximately 70 kg/m 3 to 210 kg/m 3 . Table 5.1 shows the density 
data collected over this region for Argon. The density data reveals a ± 2.76 kg/m 3 
average standard deviation of the density over these temperature and pressure ranges 
compared to a truncated virial equation of state (ref. American Institute of Physics 
Handbook). This data is compared graphically in Figure 5.27. The data shows the 
operational accuracy of this instrument to be within at worst 1% relative standard 
Carbon dioxide 

Carbon dioxide (Scott Specialty Gases, 99.999% purity) was chosen as a more 
typical, non-ideal system to test the performance of the densimeter over a broader range 



116 
temperatures. The P,T, and p data is represented in Table 5.2. The standard deviation 
is ± 10.9 kg/m 3 compared to densities calculated via a Redlich-Kwong equation of state 
that was generated with the experimental data (see reference 11). The data in Table 5.2 is 
graphically represented in Figure 5.28. Again, for the carbon dioxide data an upper limit 
seems to be approximately 1% for the relative standard deviation. 

Conclusion 

In general, the relative standard deviation that is reported for the above data would 
not be acceptable as an upper limit for operation, but this densimeter at the outset is 
designed for high pressure, non-ideal fluids where equations of state cannot begin to 
predict accurately the densities of these surface regions. Furthermore, the experimental 
relative standard deviations should decrease at higher densities if this indeed is a 
systematic deviation. 

The densities that were sampled over these temperature and pressure ranges were 
generally in agreement with the standard equations of state that were chosen for 
comparison. Experimentally, what did not compare favorably were the density values as 
the temperature deviated further from the ambient. It was apparent that the response of 
the LVDT was the source of error. It was reasoned that the LVDT transformer was 
optimized for linearity at or near ambient temperatures (although operational range of the 
LVDT was given to be 220 K - 394 K ), because there was an attempt to thermostat 
LVDT and the result was a much better agreement of density data. Thus, the obstacle 
with the current design of the instrument is the response of the LVDT as the temperature 



117 



Table 5.1. Experimental argon (39.948 g/mol) densities compared to Virial EOS 
densities. 



T(K) ± 0.01 


P(psia)± 0.5 


p exp (kg/m 3 ) ± 2.76 SD 


Pvinal (kg/m 3 ) 




563.6 


69.29 


69.66 




753.7 


94.22 


94.36 




858.5 


107.3 


108.2 


278.15 


1016.8 


128.2 


129.6 




1208.3 


153.3 


156.1 




1359.7 


172.8 


177.6 




1417.4 


183.0 


185.8 




1544.5 


198.8 


204.2 




581.6 


68.55 


69.05 




740.7 


88.70 


88.75 




887.2 


106.3 


107.2 




995.3 


120.5 


121.0 




1000.8 


120.7 


121.7 


288.15 


1150.0 


138.6 


141.0 




1253.2 


151.2 


154.6 




1270.0 


154.1 


156.9 




1416.7 


174.0 


176.5 




1471.3 


181.2 


183.7 




1671.9 


205.1 


211.4 




578.4 


66.10 


65.95 




731.0 


83.47 


83.96 




864.9 


99.65 


99.96 




1014.1 


117.2 


118.0 




1166.1 


135.0 


136.6 


298.15 


1304.5 


151.8 


153.9 




1312.7 


152.4 


154.9 




1412.0 


164.0 


167.4 




1444.4 


169.7 


171.5 




1614.5 


189.4 


193.2 




1755.8 


208.3 


211.5 



118 









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o *o *o * 


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in 


ill mLU mm 10 


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T— 


x— 


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> CM> CN> CM 


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•« 












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o 



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O 



o 
w 

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o 

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CM 



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o 
o 



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o 

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o 



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119 



Table 5.2 Experimental carbon dioxide (44.01 1 g/mol) densities compared to Redlich- 
Kwong calculated densities. 



T(K) ± 0.01 


P(psia)± 0.4 


p exp (kg/m 3 )± 10.9 SD 


Prk (kg/m 3 ) a 


290.15 


1017.5 


841.1 


830.3 




1068.8 


843.1 


837.9 




1127.9 


851.7 


846.0 




1355.3 


868.2 


873.1 




1612.4 


889.8 


898.5 


297.15 


1048.6 


770.6 


758.4 




1127.3 


783.0 


774.5 




1257.5 


804.1 


798.7 




1401.3 


821.1 


820.8 




1573.0 


839.7 


843.0 


300.15 


1005.5 


700.1 






1052.8 


720.5 


. 




1329.8 


787.0 


780.6 




1581.4 


819.0 


819.6 




1804.3 


836.3 


846.6 


303.15 


1235.4 


730.1 


723.8 




1500.5 


780.7 


780.4 




1769.3 


814.9 


819.6 


307.15 


1172.2 


584.1 


618.6 




1219.0 


639.1 


649.2 



a Values taken from reference 5 1 . 



120 



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121 
varies. This could be mitigated via the isolation of the LVDT from the temperature 
controlled environment or the replacement of the LVDT with a hardier model. 



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BIOGRAPHICAL SKETCH 
Troy Daniel Halvorsen was born on February 20, 1968. in Belvidere Illinois, to 
Dan and Susan Halvorsen. He was graduated from Southeast High School in Springfield, 
Illinois in June 1 986. He was graduated from Illinois State University in Normal, Illinois 
in May 1990 with a Bachelor of Science in Chemistry. He completed his Master's degree 
in 1993, under the guidance of Professor Cheryl Stevenson, also at Illinois State. Since 
August of 1993 he has attended Graduate School at the University of Florida where he 
has pursued the degree of Doctorate of Philosophy in physical chemistry. He enjoys 
spending time with friends and family, especially his wife, Kim, and daughter, Alana. 



125 



I certify that I have read this study and that in my opinion it conforms to the 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality 
as a dissertation for the degree of Doctor of Philosophy. 




Jofih R. Reynolds 
fofessor of Chemisuy 

I certify that I have read this study and that in my opinion it conforms to the 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality 
as a dissertation for the degree of Doctor of Philosophy. 

Timothy^. Anderson 
Professor of Chemical Engineering 
Sciences 



This dissertation was submitted to the Graduate Faculty of the College of Liberal 
Arts and Sciences and to the Graduate School and was excepted in partial fulfillment of 
the requirements for the degree of Doctor of Philosophy 



August 1998 



Dean, Graduate School 



I certify that I have read this study and that in my opinion it conforms to the 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




William Weltner. Jr., Chairman' 
Professor of Chemistry 



I certify that I have read this study and that in my opinion it conforms to the 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality 
as a dissertation for the degree of Doctor of Philosophy. 



Jz&Kul 



o 



Samuel O. Colgate 
Professor of Chemistry 




I certify that I have read this study and that in my opinion it conforms to the 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality 
as a dissertation for the degree of Doctor of Philosophy 



Willis B. Person 
Professor of Chemistry 



I certify that I have read this study and that in my opinion it conforms to the 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality 
as a dissertation for the degree of Doctor of Philosophy ' 




Marvin L. Muga 
Professor of Chemistry 






UNIVERSITY OF FLORIDA 



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